ARITHMETIC THE SECOND material career. He lived so long that although some of his •work (his theory of irrational numbers, described presently) had been familiar to all students of analysis for a generation before his death, he himself had become almost a legend and many classed him with the shadowy dead. Twelve years before his death, Teubner's Calendar for Mathematicians listed Dede- kind as having died on 4 September 1899, much to Dedekind's amusement. The day9 4 September, might possibly prove to be correct, he wrote to the editor, but the year certainly was wrong. * According to my own memorandum I passed this day in perfect health and enjoyed a very stimulating conversation on '^system and theory"' with my luncheon guest and honoured friend Georg Cantor of Halle.' Dedekind's mathematical activity impinged almost wholly on the domain of number in its widest sense. \Ve have space for only two of his greatest achievements and we shall describe first his fundamental contribution, that of the 'Dedekind cut% to the theory of irrational numbers and hence to the founda- tions of analysis. This being of the very first importance we may recall briefly the nature of the matter. If a, & are common whole numbers, the fraction a & is called a rational number; if no whole numbers m, n exist such that a certain "number' A is expressible as m/ra, then A* is called an irrational number. Thus V2, Vs, A/6 are irrational numbers. If an irrational number be expressed in the decimal notation the digits following the decimal point exhibit no regularities - there is no *period* which repeats, as in the decimal representations of a rational number, say 13/11, = 1-18181S ... , where the '18' repeats indefinitely. How then, if the representation is entirely lawless, are decimals equivalent to irrationals to be defined, let alone manipulated? Have we even any clear conception of what an irrational number is? Eudosus thought he had, and Dedekind's definition . of equality between numbers, rational or irrational, is identical with that of Eudoxus (see Chapter 2). If two rational numbers are equal, it is no doubt obvious that their square roots are equal. Thus 2x3 and 6 are equal; so also then are Vz x 3 and V&. But it is not obvious that Vi x Vs = V2 X 3, and hence that V2 X Va *= Vi". The un- 573